Monitoring magma pressure buildup at depth and transport to surface is a key point for improving volcanic eruption prediction. Seismic waves, through their velocity dependence to stress perturbations, can provide crucial information on the temporal evolution of the mechanical properties of volcanic edifices. In this article, we review past and ongoing efforts for extracting accurate information of temporal changes of seismic velocities at volcanoes continuously in time using records of ambient seismic noise. We will first introduce the general methodology for retrieving accurate seismic velocity changes from seismic noise records and discuss the origin of seismic velocity temporal changes in rocks. We will then discuss in a second part how noise-based monitoring can improve our knowledge about magmatic activity at a long (years) to a short (days) time scale taking example from Piton de la Fournaise volcano (La Réunion). We will also mention ongoing efforts for operational noise-based seismic monitoring on volcanoes. Further, we will discuss perspectives for improving the spatial localization of detected velocity changes at depth with a special focus on the use of dense seismic arrays. In the last part, we will finally explore the complex response of volcanic regions to seismic shaking with an example from Japan and show how imaging seismic velocity susceptibility allows characterizing the state of pressurized fluids in volcanic regions.